Projection

Projection

Definition of Projection

Take two vectors in $R^n$, $\vec{u}$ and $\vec{v}$.

The projection of $\vec{u}$ onto $\vec{v}$ is the vector paralell to $\vec{v}$ that’s magnitude is component of $\vec{u}$ in the direction of $\vec{v}$.

One way to visualize this is to shine a flashlight from a direction perpendicular to $\vec{v}$, with $\vec{u}$ between the flashlight and $\vec{v}$ (pretend the vectors are opaque). The shadow the flashlight casts onto $\vec{v}$ will be the projection of $\vec{u}$ onto $\vec{v}$.

The notation and formula for projection of $\vec{u}$ onto $\vec{v}$ is:

\[\text{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{|| \vec{v} ||^2} \vec{v}\]

Component

The length of the projection of $\vec{u}$ in the direction of $\vec{v}$ is called the component of $\vec{u}$ in the direction of $\vec{v}$ and is written as:

\[\text{comp}_v u = || \text{proj}_{\vec{v}} \vec{u} || = \frac{\vec{u} \cdot \vec{v}}{|| \vec{v} ||}\]

Orthogonal

The orthogonal projection of $\vec{u}$ onto $\vec{v}$ is given as:

\[\text{orth}_\vec{v} \vec{u} = \vec{u} - \text{proj}_{\vec{v}} \vec{u}\]

This is the component of $\vec{u}$ that is orthogonal to $\vec{v}$.